I am able to calculate the gain of the differential pair with the darlington pair along with the input and output resistances. however I'm having trouble understanding how the op amp works once its connected to the differential amplifier collector terminal.

From what i understand, the output resistance of the Differential pair is 3.75K (the collector resistance) and that resistance would appear at the operational amplifier.. meaning the gain will be dependent on the ratio of the op amp resistance of 5k ( R15) and the input resistance 3.75K.

However that it incorrect. what idea am i missing ? [ i know if i assume that the collector resistance is 5k . I arrive at the correct solution according to the simulation (Av=5) ]

(Please note the the written equation is not meant for this schematic.)

Note 1. A current source with a shunt resistor is the same as a voltage source with a series resistance. ( at a specific value )
Note 2. The transistor stage is a voltage to differential current converter.
Note 3. The op-amp stage is a differential voltage to voltage converter.

Let's make a few assumptions for easier calculations.

Transistor gain is high enough that B is 1 ( not .999xxx or something)
The current mirror is infinite impedance.
C5 (I don't think it is needed) and R34 are not in the equation.
calculations ignore DC values
R10 and R11 are not in the calculations
Vbe does not change as a result of signal

Here we go:

If the input is raised by 0.1V the collector of Q5 is raised by 0.05V
Q1/Q2 current increases by 50uA ( 0.05V / 1000R )
Q3/Q4 current decreases by 50uA ( 0.05V / 1000R )
( Note 2.)

50uA with a 3750R shunt can be represented by a 0.1875V source ( 50uA * 3750R ) with a 3750R series impedance. ( Note 1. )

This gives us ( differentially ) a 0.375V signal with an equivalent of 3750R in each leg.

Now it is easier to calculate the op-amp gain stage.

In the Q1/Q2/U3A+ip path the signal would get 5000 / ( 3750 + 5000 ) or 0.571429 of any signal. ( Straight voltage divider. )

In the Q3/Q4/U3A-ip path the signal gain is calculated as an inverting amplifier as 5000 / 3750 or 1.3333...